Grassmann Algebra

TheInteriorProduct.nb 23
� m = 2
�Α� 2 2
� �Α1 � Α2� ���� �Α1 � Α2� � � Α1 ���� Α1
Α2 ���� Α1
Α1 ���� Α2
�
Α2 ���� Α2
If Α is interpreted as a bivector, then �Α� is called the area of Α. �Α� is in fact the area of the
2 2 2 2
parallelogram formed by the vectors Α1 and Α2 .
Measure�Α1 � Α2�
���������������������������������������������������������������������������
��Α1 � Α2� 2 � �Α1 � Α1� �Α2 � Α2�
Graphic showing a parallelogram formed by two vectors Α1 and Α2 , and its area.
Because of the nilpotent properties of the exterior product, the measure of the bivector is
independent of the way in which it is expressed in terms of its vector factors.
Measure��Α1 �Α2��Α2�
���������������������������������������������������������������������������
��Α1 � Α2� 2 � �Α1 � Α1� �Α2 � Α2�
Graphic showing a parallelogram formed by vectors �Α1 �Α2� and Α2 , and its area.
� m = 3
�Α 3 � 2
� �Α1 � Α2 � Α3� ���� �Α1 � Α2 � Α3� �
�����
Α1 ���� Α1 Α1 ���� Α2 Α1 ���� Α3 �����
� Α2 ����
���� Α1 Α2 ���� Α2 Α2 ���� Α3 �
����
� Α3 ���� Α1 Α3 ���� Α2 Α3 ���� Α3 �
If Α 3 is interpreted as a trivector, then �Α 3 � is called the volume of Α 3 . �Α 3 � is in fact the volume
of the parallelepiped formed by the vectors Α1 , Α2 and Α3 .
V � Measure�Α1 � Α2 � Α3�
�
���Α1 � Α3� 2 �Α2 � Α2� �
2 �Α1 � Α2� �Α1 � Α3� �Α2 � Α3� � �Α1 � Α1� �Α2 � Α3� 2 �
�Α1 � Α2�2 �Α3 � Α3� � �Α1 � Α1� �Α2 � Α2� �Α3 � Α3��
Note that this expression has been simplified somewhat as permitted by the symmetry of the
scalar product.
Graphic showing a parallelepiped formed by vectors Α1, Α2 and Α3 , and its volume.
2001 4 5